This page answers the most common questions readers ask after reading our detailed article on the biased Monty Hall problem. It is designed as a simple reference for anyone who wants to understand the calculations and logic more clearly.
Q1. Why do we multiply 1/3 by 0.8 in the calculation?
Answer:
We multiply 1/3 × 0.8 because we are calculating the chance that two connected events happen together:
- The car is behind Door 1 (probability = 1/3)
- Monty opens Door 2 when he has a choice (probability = 0.8)
In probability, when one event happens after another, we multiply:
So:
This represents how often this exact situation occurs.
Q2. What does the value 0.2667 actually mean?
Answer:
The value 0.2667 means 26.67% of all games.
In simple terms:
Out of 100 similar games, about 27 times:
- The car is behind Door 1
- Monty opens Door 2
This is not the final probability. It is a “raw weight” used for comparison.
Q3. Why don’t we just say the chance is 50–50 after one door opens?
Answer:
Because Monty’s behavior is not random.
He knows where the car is and follows rules.
His choice contains information.
So the remaining doors are not equal.
Assuming 50–50 ignores this hidden information.
Q4. What does “conditional probability” mean in this problem?
Answer:
Conditional probability means:
“What is the chance of something happening, given that something else already happened?”
Example:
“What is the chance the car is behind Door 1, given that Monty opened Door 2?”
We write this as:
This is the heart of the Monty Hall problem.
Q5. Why do we remove the Door 2 = Car scenario?
Answer:
When Monty opens Door 2, it shows a goat.
So Door 2 cannot have the car.
That possibility becomes impossible and must be eliminated.
Probability must be redistributed among remaining doors.
Q6. What does “normalizing” mean?
Answer:
Normalizing means converting raw numbers into proper probabilities.
Example:
Raw values:
- Door 1 → 0.2667
- Door 3 → 0.3333
Total = 0.6
To get real probabilities, we divide each by 0.6.
This ensures the total becomes 100%.
Q7. Why do we divide by 0.6?
Answer:
Because 0.6 represents the total probability of all remaining realistic scenarios.
Dividing by 0.6 rescales the numbers so they sum to 1 (or 100%).
This keeps probability mathematically correct.
Q8. Why is switching still better even with bias?
Answer:
Because your original choice is still weak.
You picked randomly.
Bias only changes how much information Monty reveals.
It does not magically improve your first guess.
So switching remains advantageous.
Q9. Why is switching extremely powerful when Monty opens Door 3?
Answer:
Because Monty rarely opens Door 3.
When he does, it is unusual.
Unusual behavior carries strong information.
It signals that the car is likely elsewhere.
Q10. Is this problem really about Bayes’ Theorem?
Answer:
Yes.
Behind the scenes, we are using:
But we apply it using logic instead of formulas.
This makes it easier to understand.
Q11. How does this apply to business and investing?
Answer:
In real life:
- People have habits
- Decisions show patterns
- Rare actions matter most
When someone behaves unusually, it often signals hidden information.
Good decision-makers update their beliefs.
Bad ones stay emotionally attached.
Q12. Why do smart people still get this wrong?
Answer:
Because of psychology:
- Ego attachment
- Fear of changing
- Overconfidence
- Preference for consistency
Humans prefer feeling right over being right.
Probability rewards flexibility.
Q13. Is simulation better than theory?
Answer:
Simulation supports theory.
By running thousands of trials, we confirm that:
- Math matches reality
- Bias changes outcomes
- Switching remains superior
That’s why our website includes a simulator.
Q14. What is the main lesson of the biased Monty Hall problem?
Answer:
The main lesson is:
Do not judge choices in isolation.
Judge them in context.
Every decision system includes behavior.
Understanding behavior improves outcomes.
Q15. Where should beginners focus first?
Answer:
Beginners should focus on:
- Understanding why probabilities start at 1/3
- Seeing how Monty’s behavior changes things
- Learning why normalization is needed
- Practicing with simulations
Once these are clear, the rest becomes easy.
Final Note
This Q&A guide is designed to complement our main article.
For full explanations, calculations, and business applications, please refer to the main post.
Understanding probability is not about memorizing formulas.
It is about learning how information changes decisions.